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Calculus is used by taking the derivative of the log-likelihood function with respect to the parameter, setting it to zero to find critical points, and using the second derivative test to confirm that the critical point is a maximum.

Conditions: The log-likelihood function is differentiable with respect to the parameter $\theta$.; The parameter space allows for a maximum to exist.

Calculus is used by taking the derivative of the log-likelihood function with respect to the parameter, setting it to zero to find critical points, and using the second derivative test to confirm that the critical point is a maximum.

Conditions: The log-likelihood function is differentiable with respect to the parameter $\theta$.; The parameter space allows for a maximum to exist.

We use the log-likelihood because the natural logarithm is a strictly increasing (monotonic) function, so maximizing the log-likelihood yields the exact same parameter value as maximizing the original likelihood. Additionally, the logarithm converts the product of independent probabilities into a sum, which makes differentiation much easier.

Conditions: The likelihood function $L(\theta)$ is strictly positive.; The base of the logarithm is greater than 1 (e.g., the natural logarithm).

We use the log-likelihood because the natural logarithm is a strictly increasing (monotonic) function, so maximizing the log-likelihood yields the exact same parameter value as maximizing the original likelihood. Additionally, the logarithm converts the product of independent probabilities into a sum, which makes differentiation much easier.

Conditions: The likelihood function $L(\theta)$ is strictly positive.; The base of the logarithm is greater than 1 (e.g., the natural logarithm).